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Current Pharmaceutical Biotechnology, 2013, 14, 000-000
1
Measurements of Single Molecules in Solution and Live Cells Over Longer
Observation Times Than Those Currently Possible: The Meaningful Time
Zeno Földes-Papp*
Helios Clinical Center of Emergency Medicine, Department for Internal Medicine, Alte-Koelner-Strasse 9, D-51688
Koeln-Wipperfuerth, Germany
Abstract: Monitoring translational diffusion of single molecules in solution or in a living cell, particularly DNA and proteins brings valuable information unperturbed by interaction with an artificial surface. The article derives theoretical relationships for time intervals during which just one molecule in the effective probe region can be studied, the time we call
meaningful time. This time is greater than the transit time of the molecule through the detection volume, as a single molecule will likely reenter the detection volume several times during measurement. From the infinitely stretched molecular
Poisson distribution of single molecules or particles, we select the contribution of the selfsame molecule or particle by applying rules for choosing appropriate statistics for the single-molecule trajectories. The results point to a useful and sensitive predictive power of the derived relationships. The meaningful time relationships are the criteria to check the experimental single molecule data measured under conditions of normal and anomalous Brownian diffusion of the molecules of
interest. At femtomolar bulk concentration, it would be possible to observe an individual molecule over a second time interval or longer during which biological processes — and not conformational biophysical changes — are just starting.
Keywords: Single molecule, Brownian motion, translational normal diffusion, translational anomalous diffusion, solution, live
cell, theoretical relationships for time intervals during which just one molecule in the effective probe region can be studied,
meaningful time, single molecule spectroscopy, single molecule imaging.
1. INTRODUCTION
One of the reasons for exploring single molecules in solution or in a live cell is to ask whether behavior differs from
one copy of a molecule to the next copy. Each molecule experiences its own environment that in turn influences its behavior. In addition, some molecules, such as enzymes, have
different behaviors at different times, which are smeared out
if only ensemble averages are measured. For some biological
questions, single-molecule methods are not just desirable,
but a necessity. For instance, at the cellular level, gene expression is inherently a single-molecule problem. Any cell
contains at most two copies of the genetic DNA, depending
on the stage of the cell cycle. The number of copies of a particular messenger RNA is small. And many of the final
products — the proteins — are produced in small quantities,
sometimes only a single copy. However, it is difficult to detect interactions of a single molecule that diffuses freely in
solution or even in a live cell. The intention of this original
research article is to focus on physical relationships of the
time intervals during which just one molecule in the effective
probed region can be studied without immobilization
(attachment, firm binding) or hydrodynamic flow. These
characteristic time intervals we call meaningful time since
they bring significant information on the molecular processes
of just one molecule (one and the same molecule).
*Address correspondence to this author at the Helios Clinical Center of
Emergency Medicine, Department for Internal Medicine, Alte-KoelnerStrasse 9, D-51688 Koeln-Wipperfuerth, Germany;
E-mail: [email protected];
Webpage: http://publicationslist.org/Zeno.Foldes-Papp
1389-2010/13 $58.00+.00
What is a theoretical model from which we may start? In
1990, the laboratory of Richard A. Keller in Los Alamos
produced the first paper on the detection of individual molecules passing through the probe region one by one in a hydrodynamic flow with one dye molecule per second [1].
Since the publication of this paper, the area of singlemolecule detection has grown tremendously with emphasis
not just on observing single molecule signatures, but also on
applying single-molecule detection to basic chemical and
biological problems in applied and fundamental studies.
Typically, single-molecule measurements, for example with
a single enzyme molecule, are performed by absorption
(immobilization, firm binding, attachment) of the molecule
on a surface [2] or on intracellular structures [3] so that its
behavior can be observed over a period of time. Miniaturization is also having a major impact on the sensitivity of detection by applying zero-mode waveguides consisting of subwavelength holes in a metal film for parallel analysis of single-molecule dynamics at high ligand concentration (e.g.,
micromolar concentrations). Such guides can provide zeptoliter observation volumes (1 zeptoliter = 10-21 L) [4]. For
direct observation of single enzyme activity, enzymes are
absorbed (immobilized, firmly bound, attached) onto the
bottom of the waveguide in the presence of a solution containing the fluorescent-tagged ligand molecules. There are
technical hurdles associated with doing these experiments
resulting from absorption (immobilization, firm binding,
attachment). Most of the experimental single-molecule studies are combined with simulation results. In addition, theoretical and simulation methods were applied that operate
directly on the photon arrival trajectories of a single mole© 2013 Bentham Science Publishers
2 Current Pharmaceutical Biotechnology, 2013, Vol. 14, No. 3
cule by evaluating a likelihood function without having to
average over many molecules.
In the case of single molecules freely diffusing through a
solution, Brownian motion causes these molecules to be randomly transported through small (e.g., femtoliter-sized)
sampling volumes in three dimensions or in two dimensions
(cell membrane). Translational normal diffusion in solution
or translational anomalous diffusion in a live cell allows us
to monitor molecules that are not immobilized or attached
(firmly bound) to subcellular structures. As modeling of reentries is the primary purpose of this paper, the physical distinction of cases of meaningful and non-meaningful reentries
was first made in ref. [5]. For example, if the measuring signal indicates that a molecule diffuses out of the observation
volume V and right back in, it is still likely the same molecule. The number of reentries of one and the same molecule
that results in a useful burst size is meaningful and of interest. But if the molecule sits at the border of the probe region
V, crosses in and out, and therefore has many reentries,
none of those reentries results in a useful burst size. Those
reentries of one and the same molecule are non-meaningful.
The entries of new molecules of the same kind into V are
simply called ‘entries’. Ways to precisely quantify just one
and the same single molecule are in high demand.
2. THEORETICAL RESULTS AND DISCUSSION
In the very dilute solutions associated with fluctuation
spectroscopy and imaging, the molecule in the probe volume
is most probably the molecule that just diffused out, turned
around, and diffused back in, i.e. reentered. Most people
consider reentries a major problem. The challenging goal is
the theoretical and experimental demonstration of reentries
of a single molecule or particle into the probe volume [5].
For the molecular number N < 1 measured in the probe volume V, the physical meaning of N < 1 is probabilistic.
Therefore, probabilistic approaches are used in order to derive the physical relationships; the derived formulas do not
depend upon the geometry of V or of the underlying translational diffusive process (three-dimensional diffusion in
solution and in the cytoplasm/nucleoplasm of live cells, twodimensional diffusion in membranes of live cells). The correctness of the theoretical approaches and the formulas was
shown by simulation experiments based on classical meanfield approximation using the diffusion equation for ergodic
normal motion of single 24-nm and 100-nm nanospheres [6]
and for anomalous motion in live cells [7, 8].
2.1. Meaningful Time for Studying Just One Molecule in
Solution and in Live Cells
On the second-time scale where biology takes place, data
on fluorescence fluctuation traces are rare or they are not
proven to be the signature from an individual molecule or
particle. When single fluorescent molecules or single fluorescent bead particles in a liquid traverse the laser excitation
volume, fluorescent photons are generated. The photon
bursts can be analyzed for their brightness and duration, providing information on molecular size and identity, and the
trajectory of the molecule and particle, respectively.
Zeno Földes-Papp
When measuring low-concentration targets (e.g. < 1nM),
the detected fluorescence signals become digital since the
average number of molecules
C =C
(1)
is smaller than unity (<1.0) in the probe (observation) volume V [6]. With the portion of meaningful reentries p n , n ,
the meaningful time Tm during which just one molecule in
the effective probe region can be studied, is equated to
Tm =
diff
diff
1
=
=
k P(X 1; C = T ) N ,
(2)
where diff is the translational diffusion time of the single
molecule, N stands for the molecule number in the probe
volume under the condition N < 1 that corresponds to the
Poisson probability of finding a single molecule in the probe
region (arrival of a single molecule), is the detection probability of a single molecule depending upon many parameters like molecular properties of the fluorescent molecule
and instrumental parameters of the measuring device [5].
. T = C is the average number of molecules in the probe
volume V, i.e. the mean value of X. T is simply the measurement time. k is the mean value of the reentry probability
pn(t). The main difference from other single-molecule Poisson analyses in the literature is that the final expressions no
longer contain the detection probability ; it cancels out.
The same relationship (2) holds true for diff (t) with Tm becoming simply Tm = Tm (t) [7]. The above relationship (2)
for time intervals during which just one molecule in the effective probe region can be studied — the time we call
meaningful time — has particular relevance for singlemolecule analysis in experimental and theoretical photon
streams.
The meaningful-time relationship (2) considers how low
the concentration of the bulk phase should be in order to
obtain single-molecule events. Observables that have been
successfully applied to single-molecule studies are the photon counting histogram, Förster resonance energy transfer
(FRET), the lifetime of the excited state, or the emission
spectrum of a fluorescent molecule. An alternative method
based on real-time single-molecule imaging in live cells was
proposed by Seisenberger et al. 2001 [9]. The visualization
of virus trajectories projected onto the transmitted light images that were taken with an epifluorescent microscope setup
for fluorescence detection generated fluorescence spots V.
Typically, a cell was infected with 10 to 1000 virus particles/molecules. However, only if the concentration of a
molecule is small enough, each burst of emitted 'fluorescent'
photons in the probe volume V can be approximated by a
single molecule or a virus single particle.
2.2. Stretched Distribution of Other Molecules of the
Same Kind in Solution and in Live Cells
In solution or in a live cell, there might be a stretched
distribution of other molecules of the same kind for the Poisson events X = 1 molecule, X = 2 molecules,
Measurements of Single Molecules in Solution and Live Cells
Current Pharmaceutical Biotechnology, 2013, Vol. 14, No. 3
X = 3 molecules, and so forth, yielding for the molecule
number fluctuation of just one molecule in the observation
volume V
Tm (t ) =
diff (t )
c m N A V exp{ c m N A V } ,
(3)
where c m is the molar concentration of other molecules of
the same kind in the bulk phase and NA is Avogadro's number of [mol-1].
Why should we care about these findings? The physical
relationship (3) opens a clear window on the rapid advancements that are being made in cell and molecular biology. The
equation (3) takes into account the concentration effects cm
of other molecules of the same kind and the size of the probe
region V in single molecule spectroscopy and imaging. For
example, a concentration as low as 0.1 molecule per observation volume V (N=0.1) may not be small enough for singlemolecule Förster resonance energy transfer (FRET) efficiency measurements of molecules diffusing through a laser
spot V [10]. Exemplifying a translational diffusion time
diff of 26 s and a probe volume V of 0.2 femtoliter, the
single-molecule observation time Tm is 0.2435 ms given by
the meaningful-time relationship (2). Thus, N = 0.1 is small
enough for single-molecule Förster resonance energy transfer efficiency measurements of molecules with diff = 26s
diffusing through a laser spot V = 0.2 . 10-15 L for the time
interval of 0.2435 ms under the experimental conditions.
The chance that the reentering molecule is not the original molecule depends upon the waiting time for the next entry of another molecule of the same kind. The probability
density function is the way to describe this behavior [5].
dP(t t ) d
1
=
p >0 (t ) = k =
.
dt
dt
Tm
(4)
Tm corresponds to the Markov waiting time for the next
entry of a single molecule and a single particle, respectively,
that is not the original one diffusing through a laser spot V:
Proceeding along the 3D-trajectory of a single molecule or a
single particle, we show that the occurrence of nonmeaningful reentries is much more likely than that of meaningful reentries underlying the random character of
Brownian motion without systemic drift or convection [6-8].
Entries, meaningful reentries and non-meaningful reentries
had different frequency, amplitude and time distributions
under the different experimental conditions [6-8].
2.3. Estimated Errors in Measuring the Meaningful Time
for Studying Just One Molecule in Solution and in Live
Cells
Repeated measurements of a quantity always show slight
variations due to the impossibility of reproducing highly
accurate measurements exactly. The single-molecule observation time Tm is a function of directly measurable quantities
but cannot be easily measured directly. The standard deviation of these measurements is a measure of the accuracy of
the repeatability. Provided the variances var of diff (t), c m
and V are small, then
Tm
var Tm = diff
2
var diff + Tm
c
m
2
3
2
T var c m + m var V . (5)
V For diff = const and hence Tm = const ,
var Tm =
1
c N V
2
m
2
A
2
var diff +
2
diff
c N V
4
m
2
A
2
var c m +
2
diff
c N A2 V 4
2
m
var V .
(6)
The standard deviation SDTm of Tm is
SDTm = var Tm .
(7)
SDTm is in the same unit as Tm , i.e. in seconds.
Almost all the papers about single molecules diffusing
freely in solution or in a living cell at longer observation
times than in the lower millisecond range are not true single
molecule data. Those experimental data are averages over
many molecules and they do not tell us anything about one
molecule. This appears to be the situation in single molecule
biophysics without immobilization (attachment, firm binding
to subcellular structures) of molecules.
Present fluorescence fluctuation imaging and spectroscopy are not sensitive enough to decrease the bulk concentration of molecules down to the femtomolar concentration
range. At femtomolar bulk concentration, it would be possible to observe an individual molecule over a second time
interval or longer during which biological processes — and
not conformational biophysical changes — are just starting.
We were only able to go down to the lower picomolar concentration range of the dye being used [11, 12]. Of course,
that also is an issue of the photophysical properties of the
fluorescent probes used, but that is beyond the scope of this
original research article. Right now, just one thing is really
clear: there is a lot more interesting work waiting for us out
there, and it is time to “get high on single molecule biophysics” [13].
3. CONCLUSIONS
There are two different perspectives with opposite outcomes. The first outcome is that there is the concentration
effect per probe V of other molecules of the same kind in
the bulk phase for single-molecule spectroscopy and singlemolecule imaging [14]. The meaningful time Tm gives the
measurement time for studying just one molecule in solution
and in live cells as the molecule diffuses through the detection volume and subsequently reenters to transit the detection
volume again. The second outcome is that it is virtually impossible to study a specific "single molecule" under such
conditions because of the random movement of molecules in
and out of the sampling volume V in solution, as concluded
by Li et al. [15]. The different perspectives and outcomes
have to do with the burden of evidence in single-molecule
measurements and the way we are looking at the theory. All
of this is explained by the theory, which is the starting point.
CONFLICT OF INTEREST
The author confirms that this article content has no conflicts of interest.
4 Current Pharmaceutical Biotechnology, 2013, Vol. 14, No. 3
Zeno Földes-Papp
ACKNOWLEDGEMENTS
I thank Drs. Ryan M. Rich, Ignacy Gryczynski and Zygmunt Gryczynski from the Department of Molecular Biology
and Immunology, Center for Commercialization of Fluorescence Technologies, University of North Texas Health Science Center, Fort Worth, TX 76107, USA, from the Department of Cell Biology and Anatomy, University of North
Texas Health Science Center, Fort Worth, TX 76107, USA
and from the Department of Physics and Astronomy, Texas
Christian University, Fort Worth, TX 76129, USA, Ben Barbieri from ISS Inc., 1602 Newton Drive, Champaign, IL
61822, USA, Gerd Baumann, head of the Mathematics Department of the German University in Cairo, Masataka Kinjo
from the Laboratory of Molecular Cell Dynamics, Faculty of
Advanced Life Sciences, Hokkaido University, Sapporo,
Japan, Eugenia Lamont from the Medical University of
Graz, Austria, and David M. Jameson from the Department
of Cell and Molecular Biology, University of Hawai’i at
Manoa, Honolulu, USA for their comments on the manuscript.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[7]
[8]
[9]
[10]
[11]
Shera, E.B.; Seitzinger, N.K.; Davis, L.M.; Keller, R.A.; Soper,
S.A. Detection of single molecules. Chem. Phys. Lett., 1990, 174,
553-557.
Edman, L.; Földes-Papp, Z.; Wennmalm, S.; Rigler, R. The fluctuating
enzyme: a single molecule approach. Chem. Phys., 1999, 247, 11-22.
Földes-Papp, Z.; Liao, S.-C.; Barbieri, B.; Gryczynski, K.;
Luchowski, R.; Gryczynski, Z.; Gryczynski, I.; Borejdo, J.; You, T.
Single actomyosin motor interactions in skeletal muscle. Biochimica et Biophysica Acta - Molecular Cell Research, 2011, 1813
(5),
856-866.
Reference:
Available
from:
URL
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3092855/
or
http://www.iss.com/resources/pdf/publications/skeletal-muscle.pdf
Levene, M.J.; Korlach, J.; Turner, S.W.; Foquet, M.; Craighead,
H.G.; Webb, W.W. Zero-mode waveguides for single-molecule
analysis at high concentrations. Science, 2003, 299, 682-686.
Földes-Papp, Z. Fluorescence fluctuation spectroscopic approaches
to the study of a single molecule diffusing in solution and a live
cell without systemic drift or convection: a theoretical study. Curr.
Pharm. Biotechnol., 2007, 8(5), 261-273. Reference: Available
Received: December 13, 2012
[6]
Revised: January 15, 2013
Accepted: January 16, 2013
[12]
[13]
[14]
[15]
from: URL http://www.ncbi.nlm.nih.gov/pubmed/17979724 or
http://www.iss.com/resources/pdf/publications/ffs-approaches.pdf
Baumann, G.; Gryczynski, I.; Földes-Papp, Z. Anomalous behavior
in length distributions of 3D random Brownian walks and measured photon count rates within observation volumes of singlemolecule trajectories in fluorescence fluctuation microscopy. Opt.
Express, 2010, 18(17), 17883-17896. Reference: Available from:
URL http://www.opticsinfobase.org/oe/fulltext.cfm?uri=oe-18-1717883&id=204857
Földes-Papp, Z.; Baumann, G. Fluorescence molecule counting for
single-molecule studies in crowded environment of living cells
without and with broken ergodicity. Curr. Pharm. Biotechnol.,
2011, 12(5), 824-833. Reference: Available from: URL
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3195905/
Baumann, G.; Place, R.F.; Földes-Papp, Z. Meaningful interpretation of
subdiffusive measurements in living cells (crowded environment) by
fluorescence fluctuation microscopy. Curr. Pharm. Biotechnol., 2010,
11
(5),
527-543.
Reference:
Available
from:
URL
http://www.benthamdirect.org/pages/content.php?CPB/2010/00000011/
00000005/0014G.SGM
or
http://www.iss.com/resources/pdf/publications/living-cells.pdf
Seisenberger, G.; Ried, M.U.; Endreß, T.; Büning, H.; Hallek, M.;
Bräuchle, C. Real-time single-molecule imaging of the infection
pathway of an adeno-associated virus. Science, 2001, 294, 19291932.
Gopich, I.V. Concentration effects in "single-molecule" spectroscopy. J. Phys. Chem. B, 2008, 112 (19), 6214-6220.
Földes-Papp, Z.; Liao , S.-C. J.; You, T.; Terpetschnig, T.; Barbieri,
B. Confocal fluctuation spectroscopy and imaging. Curr. Pharm. Biotechnol., 2010, 11 (6). 639-653. Reference: Available from: URL
http://www.iss.com/resources/reference/publications.html
Földes-Papp, Z.; Liao, S.-C. J.; Tiefeng, Y.; Barbieri, B. Reducing
background contributions in fluorescence fluctuation time-traces
for single-molecule measurements in solution. Curr. Pharm. Biotechnol., 2009, 10 (5), 532-542. Reference: Available from: URL
http://www.iss.com/resources/pdf/publications/single-molecule.pdf
Block, S.M. Getting High on Single Molecule Biophysics. Curr.
Pharm. Biotechnol., 2009, 10(5), 464-466. Reference: Available from:
URL http://www.iss.com/resources/pdf/publications/editorial-block.pdf
Földes-Papp, Z. What it means to measure a single molecule in a solution by fluorescence fluctuation spectroscopy. Exp. Mol. Pathol., 2006,
80
(3),
209-218.
Reference:
Available
from:
URL
http://www.sciencedirect.com/science/article/pii/S0014480006000074
or http://www.iss.com/resources/pdf/publications/measuring-in-ffs.pdf
Li, X.; Hofmeister, W.; Shen, G.; Davis, L.; Daniel, C. In: Proceedings of Materials and Processes for Medical Devices (MPMD),
Sept. 23-25, USA, 2007.